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تسجيل مستخدم جديدDivision in Associative $D$-Algebra
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Aleks Kleyn
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From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $Aotimes A$-number follows. In the paper, I considered division and division with remainder. I considered also definition of prime $A$-number.
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In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $ain R$, $a<0$, then the equation has infinitely many roots. Otherwise, the
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t $a$ and $o$ is such continuous map that $$ lim_{arightarrow 0}frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $partial^{n-1} f(x)$ of order $n-1$, we define $$ partial^n f(x)circ(a_1otimes...otimes a_n) =partial(partial^{n-1} f(x)circ(a_1otimes...otimes a_{n-1}))circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=sum_{n=0}^{infty}(n!)^{-1}partial^n f(x_0)circ(x-x_0)^n $$
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